Mathematical Modelling of Breast Cancer Cells – Immun System Interaction with the Effect of Time Delay and Glucose Treatment

Abstract

The proliferation and uncontrollable growth of tumor cells during immune system interaction are aided by insufficient oxygen in the tumor microenvironment. This leads to increased glucose uptake by tumors to support energy-providing glycolysis. Hence, there is a need to investigate mathematically how glucose risk factors promote breast cancer in the dynamics of normal cells, tumor cells, and the immune system. This understanding is essential for identifying optimal treatment strategies for breast cancer. Therefore, this research proposes a mathematical model of the interaction of breast cancer cells with immune and normal cells with the impact of glucose risk factors. The model analysis provides the stability conditions for the equilibrium points and indicates the occurrence of bifurcation, considering glucose as the bifurcation parameter. Numerical results indicate uncontrollable growth of tumor and normal cells, and suppression of immune cells when the glucose is excessive which follows the theoretical findings. Additionally, the model also incorporates a time delay to account for biological processes in tumor-immune system interaction, such as cytokine secretion by immune cells targeting tumors and the production of immunosuppressive cytokines by the tumors. Biomedical evidence suggests these processes have a delayed manifestation, explaining the asymptomatic characteristic of breast cancer. The stability analysis of the delay-incorporated model reveals that the coexisting equilibrium point becomes unstable as the time delay increases, with time delay acting as the bifurcation parameter. Furthermore, Sodium-Glucose Co-Transporters 2 (SGLT-2) inhibitors are added to the delay model to explore their potential in reducing glucose’s impact on tumor growth and immune cell suppression, assessing their effectiveness against breast cancer. The analysis of the delay model with treatment is conducted using the Lyapunov function. The findings have demonstrated the efficacy of SGLT-2 inhibitors in treating breast cancer. However, to find the best drug usage conditions and prevent problems, optimal dosage is calculated with optimal control theory. This reveals a unique optimal dosage without adverse effects, where the drug ingestion rate matches the digestion rate. In conclusion, this research has demonstrated that glucose is considered a risk factor for breast cancer, and the SGLT-2 inhibitor drug may hold potential for future breast cancer therapy.